<h2>Problem 155</h2>
<div style="color:#666;font-size:80%;">19 May 2007</div><br />
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<p>An electric circuit uses exclusively identical capacitors of the same value C.
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The capacitors can be connected in series or in parallel to form sub-units, which can then be connected in series or in parallel with other capacitors or other sub-units to form larger sub-units, and so on up to a final circuit.</p>
<p>Using this simple procedure and up to <var>n</var> identical capacitors, we can make circuits having a range of different total capacitances. For example, using up to <var>n</var>=3 capacitors of 60 <img src='project/images/p_155_capsmu.gif' width='12' height='21' alt=''  style='vertical-align:middle;' />F each, we can obtain the following 7 distinct total capacitance values: </p>
<div style='text-align:center;'><img src='project/images/p_155_capacitors1.gif' width='387' height='557' alt='' /></div>
<p>If we denote by <var>D</var>(<var>n</var>) the number of distinct total capacitance values we can obtain when using up to <var>n</var> equal-valued capacitors and the simple procedure described above, we have: <var>D</var>(1)=1, <var>D</var>(2)=3, <var>D</var>(3)=7 ...</p>
<p>Find <var>D</var>(18).</p>
<p><i>Reminder :</i> When connecting capacitors C<img src="" style="display:none;" alt="_(" /><sub>1</sub><img src="" style="display:none;" alt=")" />, C<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" /> etc in parallel, the total capacitance is C<img src="" style="display:none;" alt="_(" /><sub>T</sub><img src="" style="display:none;" alt=")" />&nbsp;=&nbsp;C<img src="" style="display:none;" alt="_(" /><sub>1</sub><img src="" style="display:none;" alt=")" />&nbsp;+&nbsp;C<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" />&nbsp;+...,
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whereas when connecting them in series, the overall capacitance is given by:
<img src='project/images/p_155_capsform.gif' width='127' height='38' alt=''  style='vertical-align:middle;' /></p>

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